Relativity : On length contraction

1. Jan 16, 2007

Delzac

1. The problem statement, all variables and given/known data
The distance between the Earth and Alpha Centauri is 4.2 light years (1 light year is the distance traveled by light in one year). If astronauts could travel at v = 0.95c, then we on Earth would assume that the trip would take the astronauts 4.2 / 0.95 = 4.4 years. The astronauts however, disagree.
a) How much time passes on the astronauts clock?
b) What distance to Alpha Centauri do the astronauts measure?

2. Relevant equations
$$\Delta T = \gamma \Delta T_0$$
$$L = L_0/\gamma$$
3. The attempt at a solution

a)
$$v=0.95c \Delta T=4.4$$
Sub the values into equation you get $$\Delta T_0 = 1.37$$

b)

We have $$L_0=4.2 v=0.95c$$
Sub into equation we have L = 1.311 light years

Another Qns :
Exam smart, $$L < L_O$$ once again?
Also, is $$L_O = 4.2$$ because we are measuring the distance between the destination, and we are at rest relative to the 2 location?

2. Jan 16, 2007

Hootenanny

Staff Emeritus
Careful, you have your times mixed up! What is the definition of proper time?

3. Jan 16, 2007

Delzac

Proper time is the time interval measured in the rest frame of an event(s)

But shouldn't the time that the astronauts experience be $$\Delta T_0$$?

4. Jan 16, 2007

Delzac

And didn't T > T_0 fit nicely?

5. Jan 16, 2007

Hootenanny

Staff Emeritus
Correct!
What are the two events in question?

6. Jan 16, 2007

Delzac

Hmmm...........so there are 2 events happening concurrently? never thought of that before

Event 1 : The astronauts moving to their destination?
Event 2 : No idea

7. Jan 16, 2007

Hootenanny

Staff Emeritus

Event 1: The astronauts leave earth.
Event 2: The astronauts arrive at Alpha Centauri

Have you learnt the lorentz transformations yet?

8. Jan 16, 2007

Delzac

Hmmm......not really, but at least i know the equation when i checked up wiki.
The lecture didn't cover lorentz transformations but from the looks of it, do we need to use lorentz transformation?(guess so)

Last edited: Jan 16, 2007
9. Jan 16, 2007

Hootenanny

Staff Emeritus
Oke doke, its just I feel the Lorentz transformations are a little more intuitive than the time dilation formula. So, now you've got your two events, can you tell me the proper time?

10. Jan 16, 2007

Delzac

From the 2 events you stated, I would still say proper time = time i measured?

This is so because since proper time = measured in the rest frame of an events.
Rest frame is me measuring the time.

So this Qns is different from the "Relativity : SpaceCraft Qns" thread because what happen for "Relativity : SpaceCraft Qns" is internal(loosely speaking) events, since it happens inside the craft.

But again, if my ans is wrong why is it that this "rule", T > T_0, is seemingly not contradicted?

Usually when i sub wrong values, T > T_0 is not obeyed.

Last edited: Jan 16, 2007
11. Jan 16, 2007

Delzac

And BTW is using lorentz transformation better then using time dilation formula?

12. Jan 16, 2007

Delzac

Hmmmm come to think of it why can't the event be :

Event : the craft moved for time $$\Delta T$$?

If this is the case, then rest frame of proper time will be astronaut's times.

13. Jan 16, 2007

Hootenanny

Staff Emeritus
An event in SR is a set of spacetime coordinates (x,y,z,t), in otherwords, something happened at (x,y,z) at time t.

14. Jan 16, 2007

Delzac

So by that you are defining that event is just an instant? and not a duration like i have mention "Event : the craft moved for time T"

Therefore the above event is not an event, thus proper time isn't astronaut's time?

15. Jan 16, 2007

Hootenanny

Staff Emeritus
Yes, think of an event as a point in space and time. Therefore, we must consider two events which I outlined above.

16. Jan 16, 2007

Delzac

therefore proper time is = 4.4

using time dilation formula, i will obtain $$\Delta T=14.09years$$ which is what the astronaut experience.

17. Jan 16, 2007

Delzac

And i think part (b) of my answer also shouldn't have a problem?

18. Jan 16, 2007

Hootenanny

Staff Emeritus
My apologies, I've got this question totally backwards. You had the correct answer in your original post. My bad. I'm gona go get some sleep now :zzz: I'm really sorry to have wasted your time

19. Jan 16, 2007

Delzac

Nah its ok, when you are back can you explain what you meant by " got the question totally backwards".

How should the question be rephrase so that your working/steps are correct?

20. Jan 16, 2007

Delzac

And also explain why your argument/steps don't apply for the original question?